Schnorr trivial reals: a construction

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null ${\Sigma^0_1}A real is Martin-L?f (Schnorr) random if it does not belong to any effectively presented null (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and appears in the author’s Ph.D. thesis. A preliminary version of this paper appeared in Electronic Notes in Theoretical Computer Science     

Quantum equilibrium and the origin of absolute uncertainty

The quantum formalism is a measurement formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that particles means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by = ¦¦ ². A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.This paper is dedicated to the memory of J. S. Bell.     

用~1H-和~(13)C-NMR方法研究聚对苯二甲酸乙二酯/聚一氧基苯甲酸酯共聚酯的序列分布

测定了PET/30PHB和PET/60PHB共聚酯的高分辨~1H和~(13)C谱,并采用INEPT方法和部分弛豫FT技术确定了季碳吸收峰,对NMR谱进行了归属.应用我们先前提出的方法对共聚酯的序列分布进行了表征.结果表明,两种共聚酯的无规度B_q值均小于1,序列分布不象Jackson等认为的那样为完全无规的.此外,文中还指出,某些研究者套用Flory公式判别PET/PHB共聚酯是否为无规的不妥之处.     

Information Theory Based Evaluation of the RC4 Stream Cipher Outputs

This paper presents a criterion, based on information theory, to measure the amount of average information provided by the sequences of outputs of the RC4 on the internal state. The test statistic used is the sum of the maximum plausible estimates of the entropies H(jt|zt), corresponding to the probability distributions P(jt|zt) of the sequences of random variables (jt)t∈T and (zt)t∈T, independent, but not identically distributed, where zt are the known values of the outputs, while jt is one of the unknown elements of the internal state of the RC4. It is experimentally demonstrated that the test statistic allows for determining the most vulnerable RC4 outputs, and it is proposed to be used as a vulnerability metric for each RC4 output sequence concerning the iterative probabilistic attack.     

Correlated randomness and switching phenomena

One challenge of biology, medicine, and economics is that the systems treated by these serious scientific disciplines have no perfect metronome in time and no perfect spatial architecture—crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. Further, many of these processes and structures have the remarkable feature of “switching” from one behavior to another as if by magic. The past century has, philosophically, been concerned with placing aside the human tendency to see the universe as a fine-tuned machine. Here we will address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at some of the many spatial and temporal patterns in biology, medicine, and economics and even begin to characterize the switching phenomena that enables a system to pass from one state to another. Inspired by principles developed by A. Nihat Berker and scores of other statistical physicists in recent years, we discuss some applications of correlated randomness to understand switching phenomena in various fields. Specifically, we present evidence from experiments and from computer simulations supporting the hypothesis that water’s anomalies are related to a switching point (which is not unlike the “tipping point” immortalized by Malcolm Gladwell), and that the bubbles in economic phenomena that occur on all scales are not “outliers” (another Gladwell immortalization). Though more speculative, we support the idea of disease as arising from some kind of yet-to-be-understood complex switching phenomenon, by discussing data on selected examples, including heart disease and Alzheimer disease.     

Phase transition in spin systems with various types of fluctuations

Various types ordering processes in systems with large fluctuation are overviewed. Generally, the so-called order–disorder phase transition takes place in competition between the interaction causing the system be ordered and the entropy causing a random disturbance. Nature of the phase transition strongly depends on the type of fluctuation which is determined by the structure of the order parameter of the system. As to the critical property of phase transitions, the concept “universality of the critical phenomena” is well established. However, we still find variety of features of ordering processes. In this article, we study effects of various mechanisms which bring large fluctuation in the system, e.g., continuous symmetry of the spin in low dimensions, contradictions among interactions (frustration), randomness of the lattice, quantum fluctuations, and a long range interaction in off-lattice systems.     

混沌中的随机性分析及其在证券中的应用

本通过对混沌的整体秩序与局部随机关系的分析，讨论了当混沌时间序列的自相关函数呈负指数衰减时，我们可以提取混沌中的内在的近似随机的时间序列，并通过运用计量经济中的AR模型进行分析与模拟，模型拟合的结果很理想。由此我们得出对于一类非线性系统同样可以运用一些线性模型来作近似分析。     

The Halting Probability Omega: Irreducible Complexity in Pure Mathematics

Some G?del centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Enriques Lecture held on October 30, 2006 Received: November 2006     

Correlations of continuous random data with major world events

The interaction of consciousness and physical systems is most often discussed in theoretical terms, usually with reference to the epistemo-logical and ontological challenges of quantum theory. Less well known is a growing literature reporting experiments that examine the mind-matter relationship empirically. Here we describe data from a global network of physical random number generators that shows unexpected structure apparently associated with major world events. Arbitrary samples from the continuous, four-year data archive meet rigorous criteria for randomness, but pre-specified samples corresponding to events of broad regional or global importance show significant departures of distribution parameters from expectation. These deviations also correlate with a quantitative index of daily news intensity. Focused analyses of data recorded on September 11, 2001, show departures from random expectation in several statistics. Contextual analyses indicate that these cannot be attributed to identifiable physical interactions and may be attributable to some unidentified interaction associated with human consciousness.